arXiv 1006.48712010-06-24quant-ph

一个精确可解的三维自旋模型的拓扑序

Topological order in an exactly solvable 3D spin model

Sergey Bravyi, Bernhard Leemhuis, Barbara M. Terhal

备注：: 39 pages, 15 figures
发表信息：: Ann. Phys., Vol. 326:4, page 839 (2011)
DOI：: 10.1016/j.aop.2010.11.002

摘要：

我们研究了三维推广的复曲面代码模型介绍了最近的夏蒙。这是一个精确可解的自旋模型，在FCC晶格上具有六个量子比特的最近邻相互作用，其基空间具有拓扑量子序。我们称之为单极子的这个模型的基本激发可以在几何上描述为矩形膜的角。我们证明了，要创建一个与其他磁单极子相隔距离R的孤立磁单极子，需要一个作用于至少R^2个量子比特的算符。由两个单极子（偶极子）和四个单极子（四极子）组成的复合粒子可以被描述为弦的端点。该模型的特点是偶极型弦是刚性的，也就是说，这种弦必须与晶格的面对角线对齐。对于周期性边界条件，基空间可以编码4g个量子比特，其中g是晶格维度的最大公约数。我们描述了一套完整的逻辑运算符作用于编码的量子位在封闭的字符串和封闭的膜。

英文摘要：

We study a 3D generalization of the toric code model introduced recently by Chamon. This is an exactly solvable spin model with six-qubit nearest neighbor interactions on an FCC lattice whose ground space exhibits topological quantum order. The elementary excitations of this model which we call monopoles can be geometrically described as the corners of rectangular-shaped membranes. We prove that the creation of an isolated monopole separated from other monopoles by a distance R requires an operator acting on at least R^2 qubits. Composite particles that consist of two monopoles (dipoles) and four monopoles (quadrupoles) can be described as end-points of strings. The peculiar feature of the model is that dipole-type strings are rigid, that is, such strings must be aligned with face-diagonals of the lattice. For periodic boundary conditions the ground space can encode 4g qubits where g is the greatest common divisor of the lattice dimensions. We describe a complete set of logical operators acting on the encoded qubits in terms of closed strings and closed membranes.

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